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Quantitative Magnetic Resonance Imaging (qMRI) signal model fitting is traditionally performed via non-linear least square (NLLS) estimation. NLLS is slow and its performance can be affected by the presence of different local minima in the fitting objective function. Recently, machine learning techniques, including deep neural networks (DNNs), have been proposed as robust alternatives to NLLS. Here we present a deep learning implementation of qMRI model fitting, which uses DNNs to perform the inversion of the forward signal model. We compare two DNN training strategies, based on two alternative definitions of the loss function, since at present it is not known which definition leads to the most accurate, precise and robust parameter estimation. In strategy 1 we define the loss as the \(l^2\)-norm of tissue parameter prediction errors, while in strategy 2 as the \(l^2\)-norm of MRI signal prediction errors. We compare the two approaches on synthetic and 3T in vivo saturation inversion recovery (SIR) diffusion-weighted (DW) MRI data, using a model for joint diffusion-T1 mapping. Strategy 1 leads to lower tissue parameter root mean squared errors (RMSEs) when realistic noise distributions are considered (e.g. Rician versus Gaussian). However, strategy 2 offers lower signal reconstruction RMSE, and allows training to be performed on both synthetic and actual in vivo MRI measurements. In conclusion, for the qMRI model considered here both strategies are valid choices for DNN-based fitting. Strategy 2 is more practical, as it does not require pre-computation of reference tissue parameters, but may lead to worse parameter estimation.

The protection of private and sensitive data is an important problem of increasing interest due to the vast amount of personal data collected. Differential Privacy is arguably the most dominant approach to address privacy protection, and is currently implemented in both industry and government. In a decentralized paradigm, the sensitive information belonging to each individual will be locally transformed by a known privacy-maintaining mechanism Q. The objective of differential privacy is to allow an analyst to recover the distribution of the raw data, or some functionals of it, while only having access to the transformed data. In this work, we propose a Bayesian nonparametric methodology to perform inference on the distribution of the sensitive data, reformulating the differentially private estimation problem as a latent variable Dirichlet Process mixture model. This methodology has the advantage that it can be applied to any mechanism Q and works as a “black box” procedure, being able to estimate the distribution and functionals thereof using the same MCMC draws and with very little tuning. Also, being a fully nonparametric procedure, it requires very little assumptions on the distribution of the raw data. For the most popular mechanisms Q, like Laplace and Gaussian, we describe efficient specialized MCMC algorithms and provide theoretical guarantees. Experiments on both synthetic and real dataset show a good performance of the proposed method.